I did a quick skim and this is what I have found:
Voigts, Michael C. âThe Letters of Bernard to Monks.â In Letters of Ascent: Spiritual Direction in the Letters of Bernard of Clairvaux, 1st ed., 46â69. The Lutterworth Press, 2013. https://doi.org/10.2307/j.ctt1cgf6cn.7 (Page 48-52 discusses Bernard of Clairvaux's letter to his cousin Robert)--Other chapters in this book may be relevant as well, but I didn't give it a proper read.
Life in the Medieval Cloister by Julie Kerr has a section of chapter 3 on monks' connection to their biological families. Not a ton on letters to their families, but might be worth the read for the sources and notes.
Krawiec, Rebecca. ""From the Womb of the Church": Monastic Families." Journal of Early Christian Studies 11, no. 3 (2003): 283-307. https://dx.doi.org/10.1353/earl.2003.0050. (Again, gave it a really quick skim. It might be helpful. It might not. What I read more focuses on church fathers redefining family, but it does discuss bio family and spiritual family. The abstract of the article is: "This article presents a scrutiny of four major monastic texts (Augustine's letters to Laetus and Ecdicia; Gregory of Nyssa's Life of Macrina; and the Egyptian monk Shenoute's sermon, "On Cleaving to Profitable Things") to argue that a "profamilial" attitude existed alongside the widely-recognized "antifamilial" one. Overall I wish to argue that even within an ascetic discourse, "family discourse" could be used to transform the notion of "family" into one cohesive Christian category. Together these texts show that in late antique Christianity the spiritual and fleshly families could be linked, rather than opposed, to attain salvation for all.")
Not sure what your time period is, but outside of the medieval period we do have this letter from  Sister Kitty Witham to her mother in 1756 about an earthquake she survived.
Walker, C. (2001). âDoe not supose me a well mortifyed Nun dead to the worldâ: Letter-Writing in Early Modern English Convents. In: Daybell, J. (eds) Early Modern Womenâs Letter Writing, 1450â1700. Early Modern Literature in History. Palgrave Macmillan, London. (This article is early modern, but perhaps the book as a whole might be somewhat useful?)
EDIT: I also found this dissertation that mentions letters and nuns. Might have some tidbits or at least some jumping off points. It had a paragraph about a queen who became an abbess and would write letters to kings and bishops. (Some might be related? Idk though. Was just skimming it.)
Norris, Janice Racine. "Anglo -Saxon Religious Women Who Went to the Continent: Group Study." Order No. 3132642, State University of New York at Binghamton, 2003,
andulkaphoto
discoholicmusic
baratheonpilled
germansingergirl
tangentwithtanner
stormborns
blakekathryn
areudirty
snowghoul
yousharknotpass
